Finite Group Databasebeta

← #16777731

Subgroups — #16777731

Order 3328.

200 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13129762C212yesmaximal
2#13114911C211yesmaximal, characteristic
3C2×#88180C411yescharacteristic
4C13×#381C411yescharacteristic
5#11718872C2211yescharacteristic
6#11718305—22no—
7#11718119—22no—
8C23×C4×C13C2×C411yescharacteristic
9C2×C13×C22⋊C4—21no—
10C2×C22⋊Dic13—21no—
11C2×C22⋊Dic13D412yes—
12#88180—41no—
13#88180—41no—
14C2×C4⋊Dic13—21no—
15C2×C4×Dic13C812yescomplemented
16C22⋊#1057—42no—
17C13×#95—41no—
18#88071—41no—
19#7348—131nomaximal, Sylow
20C24×C13C22⋊C411yescharacteristic
21C22×C4×C13—21no—
22C22×C4×C13—21no—
23C22×C4×C13—42no—
24C22×C4×C13—41no—
25C22×C4×C13C2×C811yescharacteristic
26C22×C4×C13M1611yescharacteristic
27C22×Dic13—22no—
28C22×Dic13—22no—
29C2×#1057—82no—
30C13×C22⋊C4—42no—
31C22⋊Dic13—41no—
32C22⋊Dic13—41no—
33C22⋊Dic13—42no—
34C4⋊Dic13—42no—
35C4×Dic13—42no—
36#1820—131no—
37#1317—132no—
38C23×C13—22no—
39C23×C13—81no—
40C23×C13C22⋊C811yescharacteristic
41C23×C13#9911yescharacteristic
42C23×C13#10011yescharacteristic
43C2×C4×C13—21no—
44C2×C4×C13—21no—
45C2×C4×C13—42no—
46C2×C4×C13—42no—
47C2×C4×C13—41no—
48C2×C4×C13—41no—
49C2×C4×C13—81no—
50C2×C4×C13—82no—
51C2×Dic13—42no—
52C2×Dic13—42no—
53C2×Dic13—82no—
54C2×Dic13—82no—
55#1057—162no—
56C2×#117—131no—
57#381—131no—
58#381#26111yescharacteristic
59#342—262no—
60#323—262no—
61C22×C13—21no—
62C22×C13—21no—
63C22×C13—41no—
64C22×C13—41no—
65C22×C13—41no—
66C22×C13—82no—
67C22×C13#32311yescharacteristic
68C22×C13#35212yes—
69C4×C13—41no—
70C4×C13—41no—
71C4×C13—161no—
72Dic13—82no—
73Dic13—82no—
74C23×C4C2×#26111yescharacteristic
75#117—521no—
76#117—521no—
77C2×C4⋊C4—261no—
78C2×C22⋊C4—21no—
79C2×C22⋊C4—132no—
80C2×C22⋊C4—261no—
81C2×C42—132no—
82C22⋊C8—522no—
83#95—41no—
84#95—521no—
85C2×C13—21no—
86C2×C13—22no—
87C2×C13—41no—
88C2×C13—41no—
89C2×C13#131711yescharacteristic
90C24C22⋊#26111yescharacteristic
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—42no—
94C22×C4—41no—
95C22×C4—262no—
96C22×C4—262no—
97C22×C4D13⋊C811yescharacteristic
98C22×C4#611811yescharacteristic
99C2×C8—1042no—
100C4⋊C4—522no—
101C22⋊C4—42no—
102C22⋊C4—521no—
103C22⋊C4—521no—
104C22⋊C4—522no—
105C42—522no—
106C13#734811yescharacteristic, complemented, Sylow
107C23—22no—
108C23—81no—
109C23#8810711yescharacteristic
110C23#8811111yescharacteristic
111C23#8812111yescharacteristic
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—42no—
115C2×C4—42no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—81no—
119C2×C4—82no—
120C2×C4—522no—
121C2×C4—522no—
122C2×C4—1042no—
123C2×C4—1042no—
124C8—2082no—
125C22—21no—
126C22—21no—
127C22—41no—
128C22—41no—
129C22—41no—
130C22—82no—
131C22#1171823712yes—
132C22#1171828711yescharacteristic
133C4—41no—
134C4—41no—
135C4—161no—
136C4—1042no—
137C4—1042no—
138C2—21no—
139C2—22no—
140C2—41no—
141C2—41no—
142C2#1312976311yescharacteristic
subgroupG/H
centerC2#13129763
commutatorC22×C4×C13C2×C8
FrattiniC23×C4C2×#261
FittingC13×#381C4
socleC2×C13#1317
prime radicalC24×C13C22⋊C4
cosocle#11718872C22
hypercenter#381#261
hypocenterC13#7348
LCSG ▷ C22×C4×C13 ▷ C23×C13 ▷ C22×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C2 ◁ C23 ◁ C24 ◁ C23×C4 ◁ #381

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
15. C2×C4×Dic13C8semidirect
106. C13#7348semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#131297622
#131149112
#734813